Artificial Intelligence: Problem Set 4

Problem 1

Given that P(A) = a, P(B) = b P(X|A) = c, P(X|B) = d, where a+b &gt 1.
Show that P(X|A,B) is at most min(ca,db)/(a+b-1). (E.g. if
a=b=0.9 and c=d=0.1, then P(X|A,B) is at most 0.09/0.8 = 0.125.)

Problem 2

We wish to build a program that recommends movie choices to users. We decide
to use a Bayesian network that uses features of the movies and of the users.
(Note: This is not how such programs are actually constructed. We will
discuss more realistic models later in the semester.)

A small portion of this network might involve the following random
variables:

Problem 3

(Discussion)
Suppose, which is probably true, that you could set up a web site to run
this program and collect statistical data on users, and that users were
generous about filling out questionaire with a few personal questions
and extensive questions about movie preferences, both individual
and category. How would you go about selecting the random variables and
constructing the Bayesian network? How could you use the incoming data
to improve the structure of the Bayesian network, either manually or
automatically?